return map in a sentence

1. A periodic orbit of a flow is said to be hyperbolic if none of the eigenvalues of the Poincar?return map at a point on the orbit has absolute value one.
2. This is opposed to the lines bx + c . . . also if measured velocity was in fact discretised, wouldn't the return map break up into such clusters?
3. The attractor was first observed in simulations, then realized physically after Leon Chua invented the Poincar?return maps of the attractor explicitly derived by way of compositions of the eigenvectors of the 3-dimensional state space.
4. I've shown two examples of the return maps I have gotten for all my Drosophila populations ( more or less in the same form ) to the right, though the acceleration plots and the velocity plots are from different populations.
5. I was inspired by the discussion here on how to extract nonlinear order from chaos in seemingly random time series such as the logistic map by using something called a " ( Poincare ) return map " ( a little different from the Poincare map described here on Wikipedia ).
6. It's difficult to find return map in a sentence.
7. Also a reason why I argue against noise is that angular velocity's return map does not break up into discrete lines-- it's a semi-random map with a weak / moderate f ( t + 1 ) =-f ( t ) signal . talk ) 21 : 45, 2 August 2012 ( UTC)
8. But now that acceleration is close to zero, it will most likely choose acceleration values anywhere from 0 to 500 mm / s ^ 2, so it would prefer one of the two modes ( increasing or maintaining velocity ) but not the decreasing velocity mode, such that velocity will be anywhere from 7 to 43 mm / s at t = 4, but not below that, even though the velocity return map would allow for values between 0 to 7 mm / s otherwise.
9. At t = 5, it is likely the fly will decelerate ( according to the acceleration return map, because it just accelerated ) proportional to the acceleration chosen-- this locks out the increasing-velocity mode-- but because v can be anywhere from 7 to 43 mm / s ( roughly ) and acceleration anywhere from 0 to 500 mm / s ^ 2 at t = 5, we can't deduce much about t = 6 . ( The acceleration return map cuts off because of how I cropped it but the lines reach + /-500 mm / s ^ 2 .)
10. At t = 5, it is likely the fly will decelerate ( according to the acceleration return map, because it just accelerated ) proportional to the acceleration chosen-- this locks out the increasing-velocity mode-- but because v can be anywhere from 7 to 43 mm / s ( roughly ) and acceleration anywhere from 0 to 500 mm / s ^ 2 at t = 5, we can't deduce much about t = 6 . ( The acceleration return map cuts off because of how I cropped it but the lines reach + /-500 mm / s ^ 2 .)
11. I created the return maps by plotting f ( t + 1 ) on the y-axis versus f ( t ) on the x-axis, where f is either velocity or acceleration as appropriate . ( The time difference between time " t " and " t + 1 " is 1 / 15 of a second, i . e . time-series measurements are sampled at 15 Hz . ) It reveals intriguing semidiscrete stochastic decision-making on the part of Drosophila, but how do I even statistically analyse the multiple trendlines, especially so I can detect differences between genetically-different populations?
12. Of course, if v = 30 mm / s at t = 1, it's v could also be 30 or 60 mm / s at t = 2 ( with less probability ), going into loops that will be broken by drift . Walkthrough : If velocity stays the same ( at 30 mm / s ) at t = 2, acceleration is zero at t = 2, so it will likely accelerate at t = 3, in which case the velocity return map indicates ~ 60 mm / s ^ 2, in which case acceleration will be roughly + 450 mm / s ^ 2 at t = 3.
13. :: : : : : : I don't think so . . . ? ( I haven't heard of it, at least . . . ) From other papers I've seen where they looked at behavior through return maps, discrete locomotion behaviors ( trot, gallop, etc . ) would correspond to spherical'ish " clusters " on a return map, not lines . i . e . at f ( t ) there would be discrete clusters a, b, c . . . ( corresponding to the different velocity modes ), and at f ( t + 1 ) a, b, c, and the return map would simply imply switching between discrete modes.
14. :: : : : : : I don't think so . . . ? ( I haven't heard of it, at least . . . ) From other papers I've seen where they looked at behavior through return maps, discrete locomotion behaviors ( trot, gallop, etc . ) would correspond to spherical'ish " clusters " on a return map, not lines . i . e . at f ( t ) there would be discrete clusters a, b, c . . . ( corresponding to the different velocity modes ), and at f ( t + 1 ) a, b, c, and the return map would simply imply switching between discrete modes.
15. :: : : : : : I don't think so . . . ? ( I haven't heard of it, at least . . . ) From other papers I've seen where they looked at behavior through return maps, discrete locomotion behaviors ( trot, gallop, etc . ) would correspond to spherical'ish " clusters " on a return map, not lines . i . e . at f ( t ) there would be discrete clusters a, b, c . . . ( corresponding to the different velocity modes ), and at f ( t + 1 ) a, b, c, and the return map would simply imply switching between discrete modes.
16. But this implies ( according to the acceleration return map ) that acceleration is zero at t = 5, i . e . at t = 5 v is 30 mm / s ^ 2 with acceleration at t = 6 being anywhere from 0 to 500 mm / s ^ 2, but with a bias to accelerate it at + 450 mm / s ^ 2 towards 60 mm / s at t = 6 for the cycle to begin again . ( Stochastic drift, or chance of being an outlier, allows the fly to escape this cycle . ) But, if v = 60 mm / s at t = 2, then a = ~ + 450 mm / s ^ 2 at t = 2, which means a ~ =-450 mm / s ^ 2 and v ~ = 30 mm / s at t = 3, a ~ = 0 mm / s ^ 2 and v ~ = ~ 30 mm / s at t = 4 and v = 60 mm / s and a = 450 mm / s ^ 2 at t = 5-- another short-term loop which will be broken by drift.